Average Error: 11.4 → 5.7
Time: 8.0s
Precision: 64
\[\frac{a1 \cdot a2}{b1 \cdot b2}\]
↓
\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -7.352417922295524642864555630196818267831 \cdot 10^{161}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.829997385631305936289034491170991223182 \cdot 10^{-269}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.989903046627509956088269419858881704426 \cdot 10^{-222}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.722753282611232449479087576080636520303 \cdot 10^{105}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]
\frac{a1 \cdot a2}{b1 \cdot b2}
↓
\begin{array}{l}
\mathbf{if}\;a1 \cdot a2 \le -7.352417922295524642864555630196818267831 \cdot 10^{161}:\\
\;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\

\mathbf{elif}\;a1 \cdot a2 \le -2.829997385631305936289034491170991223182 \cdot 10^{-269}:\\
\;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\

\mathbf{elif}\;a1 \cdot a2 \le 1.989903046627509956088269419858881704426 \cdot 10^{-222}:\\
\;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\

\mathbf{elif}\;a1 \cdot a2 \le 1.722753282611232449479087576080636520303 \cdot 10^{105}:\\
\;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\

\mathbf{else}:\\
\;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\

\end{array}
double f(double a1, double a2, double b1, double b2) {
        double r96630 = a1;
        double r96631 = a2;
        double r96632 = r96630 * r96631;
        double r96633 = b1;
        double r96634 = b2;
        double r96635 = r96633 * r96634;
        double r96636 = r96632 / r96635;
        return r96636;
}
↓
double f(double a1, double a2, double b1, double b2) {
        double r96637 = a1;
        double r96638 = a2;
        double r96639 = r96637 * r96638;
        double r96640 = -7.352417922295525e+161;
        bool r96641 = r96639 <= r96640;
        double r96642 = b2;
        double r96643 = b1;
        double r96644 = r96643 / r96638;
        double r96645 = r96642 * r96644;
        double r96646 = r96637 / r96645;
        double r96647 = -2.829997385631306e-269;
        bool r96648 = r96639 <= r96647;
        double r96649 = 1.0;
        double r96650 = r96649 / r96643;
        double r96651 = r96639 * r96650;
        double r96652 = r96651 / r96642;
        double r96653 = 1.98990304662751e-222;
        bool r96654 = r96639 <= r96653;
        double r96655 = r96642 / r96638;
        double r96656 = r96643 * r96655;
        double r96657 = r96637 / r96656;
        double r96658 = 1.7227532826112324e+105;
        bool r96659 = r96639 <= r96658;
        double r96660 = r96639 / r96643;
        double r96661 = r96660 / r96642;
        double r96662 = r96638 / r96643;
        double r96663 = r96662 / r96642;
        double r96664 = r96637 * r96663;
        double r96665 = r96659 ? r96661 : r96664;
        double r96666 = r96654 ? r96657 : r96665;
        double r96667 = r96648 ? r96652 : r96666;
        double r96668 = r96641 ? r96646 : r96667;
        return r96668;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.4
Target11.4
Herbie5.7
\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

  1. Split input into 5 regimes

  2. if (* a1 a2) < -7.352417922295525e+161

    1. Initial program 28.1

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied associate-/r*29.2

      \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity29.2

      \[\leadsto \frac{\frac{a1 \cdot a2}{\color{blue}{1 \cdot b1}}}{b2}\]

    6. Applied times-frac16.4

      \[\leadsto \frac{\color{blue}{\frac{a1}{1} \cdot \frac{a2}{b1}}}{b2}\]

    7. Applied associate-/l*10.8

      \[\leadsto \color{blue}{\frac{\frac{a1}{1}}{\frac{b2}{\frac{a2}{b1}}}}\]

    8. Simplified10.4

      \[\leadsto \frac{\frac{a1}{1}}{\color{blue}{b2 \cdot \frac{b1}{a2}}}\]

    if -7.352417922295525e+161 < (* a1 a2) < -2.829997385631306e-269

    1. Initial program 4.9

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied associate-/r*4.9

      \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}}\]
    4. Using strategy rm

    5. Applied div-inv5.0

      \[\leadsto \frac{\color{blue}{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}}{b2}\]

    if -2.829997385631306e-269 < (* a1 a2) < 1.98990304662751e-222

    1. Initial program 16.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied times-frac3.2

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
    4. Using strategy rm

    5. Applied associate-*l/7.2

      \[\leadsto \color{blue}{\frac{a1 \cdot \frac{a2}{b2}}{b1}}\]
    6. Using strategy rm

    7. Applied associate-/l*3.3

      \[\leadsto \color{blue}{\frac{a1}{\frac{b1}{\frac{a2}{b2}}}}\]

    8. Simplified3.5

      \[\leadsto \frac{a1}{\color{blue}{b1 \cdot \frac{b2}{a2}}}\]

    if 1.98990304662751e-222 < (* a1 a2) < 1.7227532826112324e+105

    1. Initial program 4.3

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied associate-/r*3.8

      \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}}\]

    if 1.7227532826112324e+105 < (* a1 a2)

    1. Initial program 23.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
    2. Using strategy rm

    3. Applied associate-/r*22.8

      \[\leadsto \color{blue}{\frac{\frac{a1 \cdot a2}{b1}}{b2}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity22.8

      \[\leadsto \frac{\frac{a1 \cdot a2}{b1}}{\color{blue}{1 \cdot b2}}\]

    6. Applied *-un-lft-identity22.8

      \[\leadsto \frac{\frac{a1 \cdot a2}{\color{blue}{1 \cdot b1}}}{1 \cdot b2}\]

    7. Applied times-frac15.4

      \[\leadsto \frac{\color{blue}{\frac{a1}{1} \cdot \frac{a2}{b1}}}{1 \cdot b2}\]

    8. Applied times-frac12.6

      \[\leadsto \color{blue}{\frac{\frac{a1}{1}}{1} \cdot \frac{\frac{a2}{b1}}{b2}}\]

    9. Simplified12.6

      \[\leadsto \color{blue}{a1} \cdot \frac{\frac{a2}{b1}}{b2}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -7.352417922295524642864555630196818267831 \cdot 10^{161}:\\ \;\;\;\;\frac{a1}{b2 \cdot \frac{b1}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.829997385631305936289034491170991223182 \cdot 10^{-269}:\\ \;\;\;\;\frac{\left(a1 \cdot a2\right) \cdot \frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.989903046627509956088269419858881704426 \cdot 10^{-222}:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.722753282611232449479087576080636520303 \cdot 10^{105}:\\ \;\;\;\;\frac{\frac{a1 \cdot a2}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b1}}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))